The Ulam Spiral

… and many other spirals 
inspired by it …

Step (1)

The table below shows
all the whole numbers squared
in the left column
&
the respective triangular numbers
generated from these
in the right column.

Step (2)

The tabulation below shows
all the sub-totals of
the triangular numbers from
the right column in
the table in step (1).

Step (3)

The tabulation below shows
the numbers generated by the
operations described in step (2)
multiplied by 60
& the resulting products
increased by
the respective uneven numbers
starting from 1.

Step (4)

The tabulation below shows
all the sub-totals of
the numbers per (3),
resulting in all whole numbers
to the 6th. power.

Triangular Numbers Table
Base
number
Triangular
number
baseNumber
triangularNumber

0

(0 x 1)/2 = 0

1

(1 x2 )/2 = 1

4

(4 x 5)/2 = 10

9

(9 x 10)/2 = 45

16

(16 x 17)/2 = 136

Step (2)

The tabulation below shows
all the sub-totals of
the triangular numbers from
the right column in
the table in step (1).

0 = 0
0 + 1 = 1
0 + 1 + 10 = 11
0 + 1 + 10 + 45 = 56 
0 + 1 + 19 + 45 + 136 = 192




Step (3)

The tabulation below shows
the numbers generated by the
operations described in step (2)
multiplied by 60
& the resulting products
increased by
the respective uneven numbers
starting from 1.

60(0) + 1 = 1
60(1) + 3 = 63
60(11) + 5 = 665
60(56) + 7 = 3367 
60(192) + 9 = 11529




Step (4)

The tabulation below shows
all the sub-totals of
the numbers per (3),
resulting in all whole numbers
to the 6th. power.

1 = 1 = 16
1 + 63 = 64 = 26
1 + 63 + 665 = 729 = 36
1 + 63 + 665 + 3367 = 4096 = 46
1 + 63 + 665 + 3367 + 11529 = 15625 = 56




From the tabulation above
one can deduce that in general
n to the 6th. power is
the summation of triangular numbers 
per the formula shown below:

equation6thPower

6th.- power numbers derived from triangular numbers